Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis tools are based on the algebraic representation of graphs via matrices such as the graph Laplacian, or on associated graph embeddings. Such embeddings associate to each node a set of coordinates in a vector space, a representation which can then be employed for learning tasks such as the classification or alignment of the nodes of the graph. As the geometric picture provided by embedding methods enables the use of a multitude of methods developed for vector space data, embeddings have thus gained interest both from a theoretical as well as a practical perspective. Inspired by trace-optimization problems, often encountered in the analysis of graph-based data, here we present a method to derive ellipsoidal embeddings of the nodes of a graph, in which each node is assigned a set of coordinates on the surface of a hyperellipsoid. Our method may be seen as an alternative to popular spectral embedding techniques, to which it shares certain similarities we discuss. To illustrate the utility of the embedding we conduct a case study in which we analyse synthetic and real world networks with modular structure, and compare the results obtained with known methods in the literature.

翻译：基于图的模型因其在表示几乎所有类型关系数据方面的灵活性，在过去几十年取得了巨大成功。然而，虽然图本质上是组合对象，但许多重要的分析工具依赖于通过矩阵（如图拉普拉斯矩阵）对图进行代数表示，或依赖于相关的图嵌入。这类嵌入为每个节点分配一组向量空间中的坐标，这种表示可用于学习任务，例如图节点的分类或对齐。由于嵌入方法提供的几何视角使得向量空间数据开发的众多方法得以应用，嵌入在理论和实践层面都引起了广泛关注。本文受图数据分析中常见的迹优化问题启发，提出了一种推导图节点椭球嵌入的方法，其中每个节点被分配超椭球表面的一组坐标。该方法可视为流行谱嵌入技术的一种替代方案，我们讨论了二者之间的某些相似性。为说明该嵌入的实用性，我们开展了一项案例研究，分析了具有模块化结构的合成网络和真实世界网络，并将结果与文献中已知方法进行了比较。